Name: ___________________________________Bayesian Statistics, 22S:138Midterm 1, Fall 2010Show any computations that you carry out. Use the back of your exampaper if you run out of space.1. The following statements are based on the American Pet ProductsManufacturers Association 2009-2010 National Pet Owners Survey.• The probability that a randomly-selected U.S. household owns atleast one dog is 0.39.• Given that a household owns at least one dog,– the pr ob ability that it owns only 1 dog is 0.67– the pr ob ability that it owns exactly 2 dogs is 0.24(a) Given that a household owns at least one dog, what is the prob-ability that it owns 3 or more dogs? Numeric answer; show yourwork.(b) What is the p robability that a household randomly selected fromamong all U.S. household owns 3 or more dogs? (Numeric answer;show your work.)2. You wish to infer about the prop ortion p of Iowa high school fo otballcoaches who practice yoga. You decide to carry out your study inthe following way. You obtain a computerized list of all Iowa highschool football coaches. You use a computer to draw one name atrandom. You interview the first coach and ask whether h e practicesyoga. If that coach does NOT p ractice yoga, you draw another coachat random to interview. You keep on drawing names and interviewingcoaches until you get the first who says he does practice yoga. Youstop sampling after the first “yes.”1(a) You choose to use a Beta distribution as a prior for the populationproportion p. Suppose you believe that p is probably around .1and your belief is as strong as if you previously had done a studyof 20 coaches and had found 2 of them to practice yoga . Giveone appropriate choice of parameters for the Beta prior.(b) The geometric probability mass function can be us ed when theexperiment is a sequence of independent Bernoulli trials, all withthe same success probability, and the random variable of inter-est X is how many failures occur before the first success. Thegeometric pmf isf (x) =p(1 − p)x, x = 0, 1, 2, . . .0, otherwiseSuppose you interview 6 coaches who say they do not practiceyoga, and the 7th coach says he does, so x = 6. Write the likeli-hood for p based on your data.(c) Write an expression to which the posterior distribution p(p|x) isproportional.(d) Can you recognize your result in the previous problem as thekernel of a parametric density? If so, identify the parametricfamily and give the numeric values of its parameters.2NOTE: If you could not get an answer to the previous question,then answer the next three questions supposing that the posteriordistribution for p is Beta(4, 5).(e) What are the posterior mean and variance?(f) Is the Beta the conjugate prior for the geometric likelihood?Briefly justify your answer.(g) Write the R function or functions, including arguments, that youwould use to compute a 90% posterior credible set for p.(h) S uppose that the 90% posterior credible set for p tur ned ou t tobe (0.05, 0.5). What is the correct Bayesian interpretation of thisinterval? (Circle one.)i. p has different values at different times. 90% of the time, pis between 0.05 and 0.5, and 10% of the time it is not.ii. 90% of credible sets constructed using this method will con-tain the true population parameter p.iii. For a person who agreed with th e prior distribution on p,given this data the probability is .9 that the true populationproportion p is between 0.05 and 0.5.iv. none of the
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